Revisiting the $Γ(K \to e ν)/Γ(K \to μν)$ ratio in supersymmetric unified models

10.1 MB·

by  R. M. Fonseca

CFTP/12-004
PCCF RI 12-03
Revisiting the Γ(K → eν)/Γ(K → µν) ratio in
supersymmetric unified models
3 Renato M. Fonsecaa, J. C. Romãoa and A. M. Teixeirab
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n
a
J a Centro de Física Teórica de Partículas, CFTP, Instituto Superior Técnico,

5 Universidade Técnica de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
2

] b Laboratoire de Physique Corpusculaire, CNRS/IN2P3 – UMR 6533,
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p Campus des Cézeaux, 24 Av. des Landais, F-63171 Aubière Cedex, France
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e
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[

2 Abstract
v
1 It has been pointed out that supersymmetric extensions of the Standard Model can induce
1 significant changes to the theoretical prediction of the ratio Γ(K →eν)/Γ(K →µν) ≡ R ,
K
4
through lepton flavour violating couplings. In this work we carry out a full computation of all
1
one-loop corrections to the relevant ν(cid:96)H+ vertex, and discuss the new contributions to R
. K
5 arising in the context of different constrained (minimal supergravity inspired) models which
0
succeed in accounting for neutrino data, further considering the possibility of accommodating
2
a near future observation of a µ → eγ transition. We also re-evaluate the prospects for
1
: RK in the framework of unconstrained supersymmetric models. In all cases, we address the
v
questionofwhetheritispossibletosaturatethecurrentexperimentalsensitivityonR whilein
i K
X agreementwiththerecentlimitsonB-mesondecayobservables(inparticularBR(B →µ+µ−)
s
r and BR(Bu → τν)), as well as BR(τ → eγ) and available collider constraints. Our findings
a
reveal that in view of the recent bounds, and even when enhanced by effective sources of
flavourviolationintheright-handede˜−τ˜sector,constrainedsupersymmetric(seesaw)models
typically provide excessively small contributions to R . Larger contributions can be found
K
in more general settings, where the charged Higgs mass can be effectively lowered, and even
furtherenhancedintheunconstrainedMSSM.However,ouranalysisclearlyshowsthatevenin
this last case SUSY contributions to R are still unable to saturate the current experimental
K
bounds on this observable, especially due to a strong tension with the B →τν bound.
u
KEYWORDS: Supersymmetry, neutrinos, meson decays, flavour violation
1 Introduction
Neutrino oscillations have provided the first experimental manifestation of flavour violation in the
lepton sector, fuelling the need to consider extensions of the Standard Model (SM) that succeed
in explaining the smallness of neutrino masses and the observed pattern of their mixings [1–
3]. In addition to the many facilities dedicated to study neutral leptons, there is currently a
great experimental effort to search for signals of flavour violation in the charged lepton sector
(cLFV), since such an observation would provide clear evidence for the existence of new physics
beyond the SM (trivially extended to accommodate massive neutrinos). The quest for the origin
of the underlying mechanism of flavour violation in the lepton sector has been actively pursued in
recentyears, becomingevenmorechallengingastheMEGexperimentiscontinuallyimprovingthe
sensitivitytoµ → eγ decays[4], thusopeningthedoorforapossiblemeasurement(observation)in
the very near future. The current bounds on other radiative decays (i.e. (cid:96) → (cid:96) γ), or three-body
i j
decays((cid:96) → 3(cid:96) )arealreadyimpressive[5], andareexpectedtobefurtherimprovedinthefuture.
i j
Supersymmetric (SUSY) extensions of the SM offer new sources of CP and flavour violation, in
both quark and lepton sectors. Given the strong experimental constraints, especially on CP and
flavour violating observables involving the strongly interacting sector, phenomenological analyses
in general favour the so-called “flavour-blind” mechanisms of SUSY breaking, where universality
of the soft breaking terms is assumed at some high energy scale: in these constrained scenarios,
the only sources of flavour violation (FV) are the quark and charged lepton Yukawa couplings.
In order to accommodate current neutrino data, mechanisms of neutrino mass generation, such
as the seesaw (in its different realisations - for a review of the latter, see for instance [6,7]), are
often implemented in the framework of (constrained) SUSY models: in the case of the so-called
“SUSY-seesaw”, radiatively induced flavour violation in the slepton sector [8] can provide sizable
contributions to cLFV observables. The latter have been extensively studied, both at high- and
low-energies, over the past years (see e.g. [9]). Flavour violation can be also incorporated in a
more phenomenological approach, where at low-energies new sources of FV are present in the soft
SUSY breaking terms. However, these are severely constrained by a large number of observables
(see, e.g. [10] and references therein).
Inadditiontotheabovementionedrareleptondecays,leptonicandsemi-leptonicmesondecays
also offer a rich testing ground for cLFV. Here we will be particularly interested in leptonic K
decays, which (as is also the case of leptonic π decays) constitute very good probes of violation of
lepton universality. The potential of these observables, especially regarding SUSY extensions of
the SM, was firstly noticed in [11], and later investigated in greater detail in [12–14].
By themselves, these decays are heavily hampered by hadronic uncertainties and, in order to
reduce the latter (and render these decays an efficient probe of new physics), one usually considers
the ratio
Γ(K+ → e+ν[γ])
R ≡ , (1.1)
K Γ(K+ → µ+ν[γ])
sinceinthiscasethehadronicuncertaintiescanceltoaverygoodapproximation. Asaconsequence,
the SM prediction can be computed with high precision [15–17]. The most recent analysis has
provided the following value [17]:
RSM = (2.477±0.001)×10−5. (1.2)
K
Ontheexperimentalside,theNA62collaborationhasrecentlyobtainedverystringentbounds[18]:
Rexp = (2.488±0.010) ×10−5, (1.3)
K
1
which should be compared with the SM prediction (Eq. (1.2)). In order to do so, it is often useful
to introduce the following parametrisation,
RKexp = RKSM(1+∆r) , ∆r ≡ RK/RKSM−1, (1.4)
where∆r isaquantitydenotingpotentialcontributionsarisingfromscenariosofnewphysics(NP).
Comparing the theoretical SM prediction to the current bounds (i.e., Eqs. (1.2, 1.3)), one verifies
that observation is compatible with the SM (at 1σ) for
∆r = (4±4)×10−3. (1.5)
Previous analyses have investigated supersymmetric contributions to R in different frame-
K
works, as for instance low-energy SUSY extensions of the SM (i.e. the unconstrained Minimal Su-
persymmetric Standard Model (MSSM)) [11,12,14], or non-minimal grand unified models (where
higher dimensional terms contribute to fermion masses) [13]. These studies have also considered
the interplay of R with other important low-energy flavour observables, magnetic and electric
K
leptonmomentsandpotentialimplicationsforleptonicCPviolation. Distinctcomputations,based
on an approximate parametrisation of flavour violating effects - the Mass Insertion Approximation
(MIA) [19] - allowed to establish that SUSY LFV contributions can induce large contributions to
the breaking of lepton universality, as parametrised by ∆r. The dominant FV contributions are in
general associated to charged-Higgs mediated processes, being enhanced due to non-holomorphic
effects - the so-called “HRS” mechanism [20] -, and require flavour violation in the RR block of
the charged slepton mass matrix. It is important to notice that these Higgs contributions have
been known to have an impact on numerous observables, and can become especially relevant for
the large tanβ regime [20–31].
In the present work, we re-evaluate the potential of a broad class of supersymmetric extensions
of the SM to saturate the current measurement of R . Contrary to previous studies, we conduct
K
a full computation of the one-loop corrections to the ν(cid:96)H+ vertex, taking into account the im-
portant contributions from non-holormophic effective Higgs-mediated interactions. When possible
we establish a bridge between our results and approximate analytical expressions in the litera-
ture, and we stress the potential enhancements to the total SUSY contributions. In our numerical
analysis we re-investigate the prospects regarding R of a constrained MSSM onto which several
K
seesaw realisations are embedded (type I [32] and II [33], as well as the inverse seesaw [34]), also
briefly addressing L–R symmetric models [35,36]. We then consider more relaxed scenarios, such
as non-universal Higgs mass (NUHM) models at high-scale (which are known to enhance this class
of observables [13] due to potentially lighter charged Higgs boson masses), and discuss the general
prospects of unconstrained low-energy SUSY models. In all cases, we revisit the R observable
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in the light of new experimental data: in addition to LHC bounds1 on the sparticle spectrum [38]
and a number of low-energy flavour-related bounds [4,5], we implement the very recent LHCb
results concerning the BR(B → µ+µ−) [39]. As we discuss here, the increasing tension with
s
low-energy observables, in particular with B → τν, precludes sizable SUSY contributions to R
u K
even in the context of otherwise favoured candidate models as is the case of semi-constrained and
unconstrained SUSY models.
This document is organised as follows. Section 2 is devoted to the computation of the 1-loop
MSSM prediction for R . We compare our (full) result to the approximations in the literature
K
1 InournumericalanalysiswedonotrequirethelightestHiggstobeinstrictagreementwithrecentLHCsearch
results[37]: whileinthegeneralthecase(especiallyforconstrained(seesaw)models),weonlyfavourregimeswhere
itsmassislargerthan118GeV,whenconsideringsemi-constrainedandunconstrainedmodels,asignificantpartof
the studied region does indeed comply with m ∼125 GeV.
h
2
by means of the mass insertion approximation (among other simplifications), and discuss the
dominant sources of flavour violation, and the implications to other observables. Our results for a
number of models are collected in Section 3. Further discussion and concluding remarks are given
in Section 4. In the Appendices, we detail the computation of the renormalised charged lepton -
neutrino - charged Higgs vertex, and summarise the key features of two supersymmetric seesaw
realisations (types I and II) used in the numerical analysis.
2 Supersymmetric contributions to R
K
In the SM, the decay widths of pseudoscalar mesons into light leptons are given by
G2m m2 (cid:18) m2(cid:19)2
ΓSM(P±→ (cid:96)±ν) = F P (cid:96) 1− (cid:96) f2|V |2, (2.1)
8π m2 P qq(cid:48)
P
where P denotes π,K,D or B mesons, with mass m and decay constant f , and where G is
P P F
the Fermi constant, m the lepton mass and V the corresponding Cabibbo-Kobayashi-Maskawa
(cid:96) qq(cid:48)
(CKM)matrixelement. Thesedecaysarehelicitysuppressed(ascanbeseenfromthefactorm2 in
(cid:96)
Eq. (2.1)), and the prediction for their amplitude is thus hampered by the hadronic uncertainties
in the meson decay constants. As mentioned in the Introduction, ratios of these amplitudes are
independent of f to a very good approximation, and the SM prediction can then be computed
P
very precisely. Concerning the kaon decay ratio R , the SM prediction (inclusive of internal
K
bremsstrahlung radiation) is [17]
(cid:18)m (cid:19)2 (cid:32)m2 −m2(cid:33)2
RSM = e K e (1+δR ) , (2.2)
K m m2 −m2 QED
µ K µ
where δR is a small electromagnetic correction accounting for internal bremsstrahlung and
QED
structure-dependent effects (δR = (−3.60±0.04)% [17]).
QED
Figure 1: Tree level contributions to R - SM and charged Higgs.
K
In supersymmetric models, the extended Higgs sector can play an important rôle in lepton
flavour violating transitions and decays (see [20–31]). The effects of the additional Higgs are also
sizable in meson decays through a charged Higgs boson, as schematically depicted in Fig. 1. In
3
particular, for kaons, one finds [21]
Γ(K± → (cid:96)±ν) =ΓSM(K± → (cid:96)±ν)
(cid:18) m2 m (cid:19)2
× 1−tan2β K s ; (2.3)
m2 m +m
H+ s u
however, despite this new tree-level contribution, R is unaffected, as the extra factor does not
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depend on the (flavoured) leptonic part of the process.
New contributions to R only emerge at higher order: at one-loop level, there are box and
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vertex contributions, wave function renormalisation, which can be both lepton flavour conserv-
ing (LFC) and lepton flavour violating. Flavour conserving contributions arise from loop cor-
rections to the W± propagator, through heavy Higgs exchange (neutral or charged) as well as
from chargino/neutralino-sleptons (in the latter case stemming from non-universal slepton masses,
i.e., a selectron-smuon mass splitting). As concluded in [11], in the framework of SUSY models
where lepton flavour is conserved, the new contributions to ∆rSUSY are too small to be within
experimental reach.
On the other hand, Higgs mediated LFV processes are capable of providing an important
contribution when the kaon decays into a electron plus a tau-neutrino. For such LFV Higgs
couplings to arise, the leptonic doublet (L) must couple to more than one Higgs doublet. However,
at tree level in the MSSM, L can only couple to H , and therefore such LFV Higgs couplings arise
1
only at loop level, due to the generation of an effective non-holomorphic coupling between L and
H∗ - the HRS mechanism [20] - which is a crucial ingredient in enhancing the Higgs contributions
2
to LFV observables. In what follows, we address the impact of these non-holomorphic terms for
R .
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2.1 LFV Higgs mediated contributions to R
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We consider as starting point the MSSM, defined by its superpotential and soft-SUSY breaking
Lagrangian. We detail below the relevant terms for our discussion:
W = UˆcYuQˆHˆ −DˆcYdQˆHˆ −EˆcYlLˆHˆ −µHˆ Hˆ , (2.4)
2 1 1 1 2
V =−L = (M ψ ψ +h.c.)+m2 H∗H
soft soft α α α Hi i i
+(BH H +h.c.)+(cid:96)˜∗ m2 (cid:96)˜ +(cid:96)˜∗ m2(cid:96)˜
1 2 L L˜ L R R˜ R
+ (H (cid:96)˜∗ Al(cid:96)˜ +h.c.)+..., (2.5)
1 R L
where M denotes the soft-gaugino mass terms, “...” stand for the squark terms, and we have
α
omitted flavour indices. For the SU(2) superfield products, we adopt the convention Hˆ Hˆ ≡
1 2
Hˆ1Hˆ2−Hˆ2Hˆ1 (and likewise for similar cases).
1 2 1 2
From an effective theory approach, the HRS mechanism can be accounted for by additional
terms, corresponding to the higher-order corrections to the Higgs-neutrino-charged lepton interac-
tion (schematically depicted in Fig. 2).
At tree-level, the Lagrangian describing the ν(cid:96)H± interaction is given by
LH± =ν Yl†(cid:96) H−∗+h.c.
0 L R 1
(cid:16) (cid:17)
= 23/4G1/2 tanβν Ml(cid:96) H++h.c., (2.6)
F L R
4
Figure 2: Corrections to the ν(cid:96)H+ vertex, as discussed in the text.
with Ml = diag(m ,m ,m ). At loop level, two new terms are generated: ν ∆+(cid:96) H+ −
e µ τ L R 2
(cid:96) ∆0(cid:96) H0 +h.c.. The second one, with ∆0, forces a redefinition of the charged lepton Yukawa
L R 2
couplings, Yl† = Ml → Yl† ≈ Ml −∆0tanβ, which in turn implies a redefinition of the charged
v1 v1
lepton propagator; the term with ∆+ corrects the Higgs-neutrino-charged lepton vertex2. Once
these terms are taken into account, the interaction Lagrangian, Eq. (2.6), becomes
(cid:16) (cid:17)
LH± = 23/4G1/2 tanβ ν Ml(cid:96) H+
F L R
+ cosβ ν (cid:0)∆+−∆0 tan2β(cid:1) (cid:96) H++h.c.. (2.7)
L R
Since in the SU(2) -preserving limit ∆+ = ∆0, it is reasonable to assume that, after electroweak
L
(EW) symmetry breaking, both terms remain approximately of the same order of magnitude.
Hence, it is clear that the contribution associated with ∆0 (the loop contribution to the charged
lepton mass term) will be enhanced by a a factor of tan2β when compared to the one associ-
ated with ∆+. This simple discussion allows to understand the origin of the dominant SUSY
contribution3 to R .
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As we proceed to discuss, a quantitative assessment of the corrections to ∆+ and ∆0 requires
considering the higher-order effects on the vertex ν ZH(cid:96) H+ (see also [41]). The ZH matrix
L R
depends on the following (loop-induced) quantities:
• η(cid:96) and ην (corrections to the kinetic terms of (cid:96) and ν );
L L L L
• η(cid:96) (correction to the charged lepton mass term);
m
• ηH (correction to the ν(cid:96)H vertex).
The expressions for the distinct η-parameters can be found in Appendix A. Instead of ZH, which
includes both tree and loop level effects, it proves to be more convenient to use the following
combination,
tanβ (cid:18) m (cid:19)2 m (cid:16) (cid:17)−1
− K s ZH Ml ≡ (cid:15)1+∆, (2.8)
23/4G1/2 mH+ ms+mu
F
where
(cid:18) m (cid:19)2 m
(cid:15) = −tan2β K s , (2.9)
m m +m
H+ s u
(cid:34) (cid:32) (cid:33) (cid:35)
η(cid:96) ην ηH (cid:16) (cid:17)−1
∆ = (cid:15) L − L + −η(cid:96) Ml . (2.10)
2 2 23/4G1/2tanβ m
F
2An extensive discussion on the radiatively induced couplings which are at the origin of the HRS effect can be
found in [40].
3Thereareadditionalcorrectionstotheqq(cid:48)H± vertex,whicharemainlyduetoasimilarmodificationofthethe
quark Yukawa couplings - especially that of the strange quarks. This amounts to a small multiplicative effect on
∆r which we will not discuss here (see [14] for details).
5
In the above, (cid:15) encodes the tree level Higgs mediated amplitude (which does not change the SM
predictionforR ), while∆, amatrixinleptonflavourspace, encodesthe1-loopeffects. Themain
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contribution is expected to arise from η(cid:96) .
m
The ∆r observable is then related to (cid:15) and ∆ as follows:
(cid:104)(cid:16) (cid:17)(cid:16) (cid:17)(cid:105)
1+ ∆† 1+ ∆
∆r ≡ RK −1 = 1+(cid:15) 1+(cid:15) ee −1. (2.11)
RKSM (cid:104)(cid:16)1+ ∆† (cid:17)(cid:16)1+ ∆ (cid:17)(cid:105)
1+(cid:15) 1+(cid:15)
µµ
If the slepton mixing is sufficiently large, this expression can be approximated as
∆r ≈ 2 Re(∆ ) + (∆†∆) . (2.12)
ee ee
In the above, the first (linear) term on the right hand-side is due to an interference with the SM
process, and is thus lepton flavour conserving. As shown in [11], this contribution can be enhanced
through both large RR and LL slepton mixing. On the other hand, the quadratic term (∆†∆)
ee
can be augmented mainly through a large LFV contribution from ∆ , which can only be obtained
τe
in the presence of significant RR slepton mixing.
2.2 Generating ∆r: sources of flavour violation and experimental constraints
In order to understand the dependence of ∆r on the SUSY parameters, and the origin of the
dominantcontributionstothisobservable,anapproximateexpressionfor∆isrequired. Firstly, we
noticethatthepreviousdiscussion,leadingtoEq.(2.7),suggeststhattheη(cid:96) termisresponsiblefor
m
the dominant contributions to ∆r. Thus, in what follows, and for the purpose of obtaining simple
analytical expressions, we shall neglect the contributions of the other terms (although these are
included in the numerical analysis of Section 3). A fairly simple analytical insight can be obtained
when working in the limit in which the virtual particles in the loops (sleptons and gauginos) are
assumed to have similar masses, so that their relative mass splittings are indeed small. In this
limit, one can Taylor-expand the loop functions entering η(cid:96) (see Appendix A); working to third
m
orderinthisexpansion, andkeepingonlythetermsenhancedbyafactorofm tanβ mSUSY (where
τ mEW
m , m denote the SUSY breaking scale and EW scale, respectively), we obtain
SUSY EW
(cid:34) (cid:32) (cid:33) (cid:35)2
9 δ (cid:16) (cid:17)
∆r ∼ 1+X 1− m2 −1
10m2 L˜ eτ
(cid:96)(cid:101),χ0
(cid:34) (cid:32) (cid:33)(cid:35)2
3 µ2+2M2
+X2 −µ2+δ 3− 1 , (2.13)
10 m2
(cid:96)(cid:101),χ0
whereµ,M and(m2) denotethelow-energyvaluesoftheHiggsbilinearterm,binosoft-breaking
1 L˜ eτ
mass,andoff-diagonalentryofthesoft-breakingleft-handedsleptonmassmatrix,respectively. We
havealsointroducedm2 = 1((cid:104)m2(cid:105)+(cid:104)m2 (cid:105)),theaveragemasssquaredofsleptonsandneutralinos
(cid:96)(cid:101),χ0 2 (cid:96)(cid:101) χ0
(≈ m2 ), and δ = 1((cid:104)m2(cid:105)−(cid:104)m2 (cid:105)), the corresponding splitting. The quantity X is given by
SUSY 2 (cid:96)(cid:101) χ0
(cid:16) (cid:17)
X ≡ 1 m2 g(cid:48)2µM tan3β mτ m2R˜ τe , (2.14)
192π2 K 1 m2 m (m2 )3
H+ e (cid:96)(cid:101),χ0
and it illustrates in a transparent (albeit approximate) way the origin of the terms contributing
to the enhancement of R : in addition to the factor tan3β/m2 , usually associated with Higgs
K H+
6
exchanges, the crucial flavour violating source emerges from the off-diagonal (τe) entry of the
right-handed slepton soft-breaking mass matrix.
Using the above analytical approximation, one easily recovers the results in the literature,
usually obtained using the MIA. For instance, Eq. (11) of Ref. [11] amounts to
(cid:16) (cid:17) (cid:16) (cid:17)2
∆r ∼ 2X m2 + X2 m2 + X2δ2, (2.15)
L˜ eτ L˜ eτ
which stems from having kept the dominant (crucial) second and third order contributions in the
(cid:16) (cid:17) (cid:16) (cid:17)2
expansion: X2δ2 and 2X m2 +X2 m2 , respectively.
L˜ eτ L˜ eτ
Regardlessoftheapproximationconsidered,itisthusclearthattheLFVeffectsonkaondecays
into a eν or µν pair can be enhanced in the large tanβ regime (especially in the presence of low
(cid:16) (cid:17)
values of m ), and via a large RR slepton mixing m2 . Although the latter is indeed the
H+ R˜ τe
privilegedsource,noticethat,ascanbeseenfromEq.(2.15),astrongenhancementcanbeobtained
(cid:16) (cid:17)
from sizable flavour violating entries of the left-handed slepton soft-breaking mass, m2 . This
L˜ eτ
is in fact a globally flavour conserving effect (which can also account for negative contributions to
R ). Previous experimental measurements of R appeared to favour values smaller than the SM
K K
theoretical estimation, thus motivating the study of regimes leading to negative values of ∆r [11],
but these regimes have now become disfavoured in view of the present bounds, Eq. (1.5).
Clearly, these Higgs mediated exchanges, as well as the FV terms at the origin of the strong
enhancement to R , will have an impact on a number of other low-energy observables, as can be
K
easily inferred from the structure of Eqs. (2.13-2.15). This has been extensively addressed in the
literature [11–14], and here we will only briefly discuss the most relevant observables: electroweak
precision data on the anomalous electric and magnetic moments of the electron, as well as the
naturalness of the electron mass, directly constrain the η(cid:96) corrections (and η(cid:96), ηH); low-energy
m L
cLFV observables, such as τ → (cid:96)γ and τ → 3(cid:96) decays are also extremely sensitive probes of Higgs
mediated exchanges, and in the case of τ −e transitions, depend on the same flavour violating
entries. It has been suggested that positive and negative values of ∆r can be of the order of
1%, still in agreement with data on the electron’s electric dipole moment and on τ → (cid:96)γ [11–13].
Finally, other meson decays, such as B → (cid:96)(cid:96) (and B → (cid:96)ν), exhibit a similar dependence on tanβ,
tannβ/m 4 [42] (n ranging from 2 to 6, depending on the other SUSY parameters), and may
H+
also lead to indirect bounds on ∆r. In particular, the strict bounds on BR(B → τν) [5] and
u
the very recent limits on BR(B → µ+µ−) [39] might severely constrain the allowed regions in
s
SUSY parameter space for large tanβ. Although we will come to this issue in greater detail when
discussing the numerical results, it is clear that the similar nature of the K+ → (cid:96)ν and B → τν
u
processes (easily inferred from a generalization of Eq. (2.3), see e.g. [21,43]) will lead to a tension
when light charged Higgs masses are considered to saturate the bounds on R .
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Supersymmetric models of neutrino mass generation (such as the SUSY seesaw) naturally
induce sizable cLFV contributions, via radiatively generated off-diagonal terms in the LL (and to
a lesser extent LR) slepton soft-breaking mass matrices [8]. In addition to explaining neutrino
masses and mixings, such models can also easily account for values of BR(µ → eγ), within the
reach of the MEG experiment. In view of the recent confirmation of a large value for the Chooz
angle (θ ∼ 8.8◦) [3] and on the impact it might have on (m2) , in the numerical analysis of the
13 L˜ eτ
following section we will also consider different realisations of the SUSY seesaw (type I [32], II [33]
and inverse [34]), embedded in the framework of constrained SUSY models. We will also revisit
semi-constrained scenarios allowing for light values of m , re-evaluating the predictions for R
H+ K
7
under a full, one loop-computation, and in view of recent experimental data. Finally, we confront
these (semi-)constrained scenarios with general, low-energy realisations, of the MSSM.
3 Prospects for R : unified vs unconstrained SUSY models
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In this section we evaluate the SUSY contributions to R , with the results obtained via the full
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expressions for ∆r, as described in Section 2. These were implemented into the SPheno public
code [44], which was accordingly modified to allow the different studies. It is important to stress
thatalthoughsomeapproximationshavestillbeendone(aspreviouslydiscussed),theresultsbased
on the present computation strongly improve upon those so far reported in the literature (mostly
obtained using the MIA). Although the different contributions cannot be easily disentangled due
to having carried a full computation, our results automatically include all one-loop lepton flavour
violatingandleptonflavourconservingcontributions(inassociationwithchargedHiggsmediation,
see footnote 3). As mentioned before, we evaluate R in the framework of constrained, semi-
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constrained (NUHM) and unconstrained SUSY models. Concerning the first two, we assume
some flavour blind mechanism of SUSY breaking (for instance minimal supergravity (mSUGRA)
inspired), so that the soft breaking parameters obey universality conditions at some high-energy
scale, which we choose to be the gauge coupling unification scale M ∼ 1016 GeV,
GUT
(cid:16) (cid:17)2 (cid:0) (cid:1)2 (cid:0) (cid:1)2 (cid:0) (cid:1)2 (cid:0) (cid:1)2
m = m = m = m = m
Q˜ U˜ ij D˜ ij L˜ ij R˜ ij
ij
=m2δ ,
0 ij
(cid:16) (cid:17)
Al =A (Yl) . (3.1)
0 ij
ij
In the above, m and A are the universal scalar soft-breaking mass and trilinear couplings of
0 0
the cMSSM, and i,j denote lepton flavour indices (i,j = 1,2,3). In the latter case, the gaugino
masses are also assumed to be universal, their common value being denoted by M . We will
1/2
also consider the supersymmetrisation of several mechanisms for neutrino mass generation. More
specifically, we have considered the type I and type II SUSY seesaw (as detailed in Appendix B).
We briefly comment on the inverse SUSY seesaw, and discuss a L−R model.
The strict universality boundary conditions of Eqs. (3.1) will be relaxed for the Higgs sector
when we address NUHM scenarios, so that in the latter case we will have
m2 (cid:54)= m2 (cid:54)= m2. (3.2)
H1 H2 0
All the above universality hypothesis will be further relaxed when, for completeness, and to allow
a final comparison with previous analyses, we address the low-energy unconstrained MSSM.
In our numerical analysis, we took into account LHC bounds on the SUSY spectrum [38], as
well as the constraints from low-energy flavour dedicated experiments [5], and neutrino data [1,2].
In particular, concerning lepton flavour violation, we have considered [4,5]:
BR(τ → eγ) < 3.3×10−8 (90%C.L.), (3.3)
BR(τ → 3e) < 2.7×10−8 (90%C.L.), (3.4)
BR(µ → eγ) < 2.4×10−12 (90%C.L.), (3.5)
BR(B → τν) > 9.7×10−5 (2σ). (3.6)
u
8
m (GeV) M (GeV) tanβ A (GeV) sign(µ)
0 1/2 0
10.3.1 300 450 10 0 1
P20 330 500 20 -500 1
P30 330 500 30 -500 1
40.1.1 330 500 40 -500 1
40.3.1 1000 350 40 -500 1
Table 1: cMSSM (benchmark) points used in the numerical analysis.
Also relevant are the recent LHCb bounds [39]
BR(B → µ+µ−) < 4.5×10−9 (95%C.L.), (3.7)
s
BR(B → µ+µ−) < 1.03×10−9 (95%C.L.). (3.8)
When addressing models for neutrino mass generation, we take the following (best-fit) values
for the neutrino mixing angles [2] (where θ is already in good agreement with the most recent
13
results from [3]),
sin2θ = 0.312+0.017, sin2θ = 0.52+0.06,
12 −0.015 23 −0.07
sin2θ ≈ 0.013+0.007, (3.9)
13 −0.005
∆m2 = (7.59+0.20) ×10−5 eV2, (3.10)
12 −0.18
∆m2 = (2.50+0.09) ×10−3 eV2. (3.11)
13 −0.16
Regarding the leptonic mixing matrix (U ) we adopt the standard parametrisation. In the
MNS
present analysis, all CP violating phases are set to zero4.
3.1 mSUGRA inspired scenarios: cMSSM and the SUSY seesaw
We begin by re-evaluating, through a full computation of the one-loop corrections, the maximal
amount of supersymmetric contributions to R in constrained SUSY scenarios. For a first evalua-
K
tion of R , we consider different cMSSM (mSUGRA-like) points, defined in Table 1. Among them
K
are several cMSSM benchmark points from [45], representative of low and large tanβ regimes,
as well as some variations. Notice that, as mentioned before, these choices are compatible with
having a Higgs boson mass above 118 GeV but will be excluded once we require m to lie close to
h
125 GeV as suggested by LHC results [37].
As could be expected from Eqs. (2.13-2.15), in a strict cMSSM scenario (in agreement with
the experimental bounds above referred to) the SUSY contributions to R are extremely small;
K
motivated by the need to accommodate neutrino data, and at the same time accounting for values
of BR(µ → eγ) within MEG reach, we implement type I and type II seesaws in mSUGRA-
inspiredmodels(seeAppendicesB.1andB.2). Regardingtheheavy-scalemediators,weconsidered
degenerate right-handed neutrinos, as well as degenerate scalar triplets. We set the seesaw scale
aiming at maximising the (low-energy) non-diagonal entries of the soft-breaking slepton mass
matrices, while still in agreement with the current low-energy bounds (see Eqs. (3.3-3.8)). In
particular, we have tried to maximise the LL contributions to ∆r, i.e., (m2) , and to obtain
L˜ eτ
4We will assume that we are in a strictly CP conserving framework, where all terms are taken to be real. This
implies that there will be no contributions to observables such as electric dipole moments, or CP asymmetries.
9